62 A. T. MASTERMAN [januaey 



Perhaps the most recent example of this type, in the highest of 

 animals, is the universal tendency to right or left-handedness in man. 



This fourth type of symmetry must be carefully distinguished from 

 the cases of peculiar arrangement of parts due to the superposition of 

 one type of symmetry upon another, or even of one type upon itself 

 but in a different relation to the organism. Thus, in the majority of 

 the Echinoderma the " plane " type has probably been impressed upon 

 the " axial " of the Coelentera, only to be in its turn " covered " by the 

 axial type of the sedentary period. Lastly, some species, such as 

 Spatangus, have early traces of a " plane " type again superposed upon 

 the " axial," and a further reversion to the " axial " is found in Pclago- 

 thuria. Again, in the Pleuronectida? the so-called asymmetry is 

 evidently due to a second piano-symmetry superposed upon the first, 

 with the two planes of symmetry perpendicular to each other. In all 

 these cases the transition will involve an apparent asymmetry or stereo- 

 symmetry. 



The four types of symmetry and their sub-types may be shown dia- 

 grammatically as in Pigs. 1-6, and Pig. 8, in which each of the three dimen- 

 sional axes is indicated, 

 ' z: y^ J*- and their resemblances 



/ \) A - / 3-") A ' / >^ C * / ^1 ^ or differences are shown 



^_\/ /- V /^ \l by letters. The increas- 



A. a. A. A. D. • heterogeneity as the 



Fig. 10. Fig. 11. Fig. 12. °. o J 



various types are passed 



Fig. 9. — Centro-symmetry. (4a.) ] . A 



Fig. 10.— Axo-symnietry. (3a + b.) through IS shown clearly 



Fig. 11.— Piano-symmetry. (2a + b + c.) by t j ie f ormu l ae below 



Fig. 12.— Stereo-symmetry, (a + b + c + d.) j 



the figures. Again, 11 a 

 regular tetrahedron be taken instead of an octagon, because it is the 

 simplest possible solid figure, then the four main types of symmetry 

 are indicated as in Pigs. 9-12, whilst in this case the sub-types can 

 only be shown by a reference to the planes of symmetry of a tetra- 

 hedron, into which it is not necessary to enter. 



The term " asymmetry " should be strictly confined to the absence 

 of all definite " arrangement," and in this sense it has been used by 

 Spencer. 



The " asymmetry " of the chemists and physicists is, however, 

 exactly analogous to stereo-symmetry as here defined, and both pheno- 

 mena are alike due to the highest possible " arrangement " of the mass 

 in each case. Pasteur's conclusions with regard to the intimate con- 

 nection of molecular stereo-symmetry and organic phenomena have been 

 recently brought forward with renewed emphasis by Professor Japp. 1 

 The ascertained fact that the molecular stereo-symmetry is a characteristic 

 feature of " organised " matter, pointed out by him, is interesting in 

 view of the fact that the evolution of symmetry in the animal kingdom 

 indicates a gradual transference of stereo-symmetry from the molecule 



1 President's Address, Chemical Section, British Association, Bristol 189S. 



A. A 



Fig. 9. 



